Sergey Ryvkin Eduardo Palomar Lever Sliding Mode Control for Synchronous Electric Drives 2012

8/12/2019 Sergey Ryvkin Eduardo Palomar Lever Sliding Mode Control for Synchronous Electric Drives 2012

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Sliding Mode Control for SynchronousElectric Drives


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Sliding Mode Control forSynchronous Electric Drives

Sergey RyvkinRussian Academy of Sciences,

Trapeznikov Institute of Control Sciences, Moscow, Russia

Eduardo Palomar LeverUniversity of Guadalajara, CUCEI, Guadalajara, Mxico


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CR C Press/Ba lkema is a n imprin t of th e Ta ylor & Fra n cis Grou p,an informa business

2012 Taylor & Francis Group, London, UK

Typeset by MPS Limited, a Macmillan Company, Chennai, IndiaPrinted and bound in Great Britain by Antony Rowe(A CPI-group Company), Chippenham,Wiltshire

All rights reserved. No part of this publication or the informationcontained herein may be reproduced, stored in a retrieval system,or transmitted in any form or by any means, electronic, mechanical,by photocopying, recording or otherwise, without written priorpermission from the publishers.

Although al l care is taken to ensure integrity and the quality of thispublication and the information herein, no responsibility isassumed by the publishers nor the author for any damage to theproperty or persons as a result of operation or use of thispublication and/or the information contained herein.

Library of Congress Cataloging-in-Publication Data

Sliding mode control for synchronous electric drives / Sergey Ryvkin,Eduardo Palomar Lever.

p. cm.Includes bibliographical references and index.ISBN 978-0-415-69038-6 (hardback : alk. paper) 1. Electric motors,

SynchronousAutomatic control. 2. Electric motorsElectronic control.3. S l iding mode control . I . Ryvkin, Sergey I I . Palomar Lever, Eduardo.

TK2787.S65 2012621.46dc23

2011034523

Published by: CRC Press/BalkemaP.O. Box 447, 2300 AK Leiden,The Netherlandse-mail: [email protected] www.taylorandfrancis.co.uk www.balkema.nl

ISBN: 978-0-415-69038-6 (Hbk)ISBN: 978-0-203-18138-6 (eBook)


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Contents

Preface ixAcknowledgements and Dedications xiAbout the authors xiii

Introduction 1

1 Problem statement 11

1.1 Mathematical models of the drive elements 111.1.1 Synchronous motors 111.1.2 Semiconductor power converters 15

1.2 Drive control problems and their existing solutions 23

2 Sliding mode in nonlinear dynamic systems 27

2.1 Plant features and sliding mode design 272.1.1 Sufficient existence conditions of a sliding mode 30

2.2 Sufficient existence conditions of sliding mode in systemswith redundant control 33

2.3 Sliding mode design 38

3 State vector estimation 43

3.1 Information aspects of sliding mode design 433.2 Use of an asymptotical observer of the state variables 433.3 Nonlinear sliding mode observer 453.4 Physical significance of equivalent control 51

4 Synchronous drive control design 53

4.1 Single-loop control design 53

4.1.1 The two step decomposition approach 534.1.2 First step design of fictitious discontinuous control 544.1.3 Second step phase voltage control design 63

4.2 Cascade (subordinated) control 714.3 Static modes optimization 79


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vi Contents

4.3.1 Problem statement 794.3.2 Keeping maximum efficiency and minimum stator current 804.3.3 Keeping cos=1 814.3.4 Realization of the offered dependencies 84

4.3.5 Using control taskid z=0 87

5 Multidimensional switching regularization 89

5.1 Features of real sliding mode 895.2 Switching loss minimized control for VSI 90

5.2.1 Analysis of PWM laws 905.2.2 Comparative analysis of switching laws

from the switching losses viewpoint 935.2.3 Comparing PWM switching laws Numerical results 96

5.2.4 Switching loss minimizing PWM 975.3 Optimal switching losses in real sliding mode 985.4 Switching regularization of discontinuous control vector

components 1025.4.1 Control vector 1025.4.2 Simplified control 1085.4.3 Follow-up current vector control structure 1085.4.4 Test simulation of a follow-up loop 109

6 Mechanical coordinates observers 113

6.1 General formulation of the observation problem 1136.2 Observer design for permanent magnet salient-pole synchronous

motor with constant magnets 1146.2.1 Rotating coordinate system 1146.2.2 Motionless coordinate system (,) 1186.2.3 The simplified observer 120

6.3 Observer design for the synchronous reluctance motor 1216.3.1 Rotating coordinate system 121

6.3.2 The simplified observer 127

7 Digital control 129

7.1 Main principles of digital control 1297.1.1 Features of digital control 1297.1.2 Digital sliding mode 130

7.2 Digital control design for the synchronous motor 1317.2.1 Synchronous motor difference equations 1317.2.2 Angular speed control 134

7.3 Digital drive mechanical variable estimation 1367.3.1 Problem statement 1367.3.2 Permanent magnetnonsalient-polesynchronous motor

state observer 1367.3.3 The filter-observer of mechanical variables 139


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Contents vii

7.4 Parameter identification of linear digital system withvariable factors and the limited memory depth 1417.4.1 Statement of a parameter identification problem 1417.4.2 Identification condition of matrix factors 142

7.4.3 Identification of physical parameters 1447.4.4 Moment of inertia identification 144

7.5 The reference rate limiter 1457.5.1 The general problem statement 145

7.6 Reference rate limiter 1467.7 Digital control design for the electric drive with elastic connections 149

7.7.1 Control problem statement 1497.7.2 Elastic mechanical movement difference model 1527.7.3 Digital control design of elastic oscillations 154

7.7.4 State variable observer 155

8 Practical examples of drive control 159

8.1 High speed synchronous drive sensorless control 1598.1.1 Features of the control system 1598.1.2 Simulation model 1628.1.3 Drive rating parameters 1658.1.4 Sensitivity research to parameter variation 1688.1.5 Influence of A/D converter discreteness on current

measurements 1728.1.6 The influence of VSI dead time 1748.1.7 Conclusions on the simulation 175

8.2 Digital control system of the electric drive withelastic mechanical connections 1768.2.1 Control plant features 1768.2.2 Main principles of control design 1788.2.3 Dry friction and backslash compensation 1798.2.4 Closed loop simulation 181

Index 185


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Preface

This book was conceived for specialists and postgraduate students who work in thefield of electrical drive control. It is also ideal for the specialists in control theoryapplication, and in particular, sliding mode using for control of electrical motors andpower converters. The authors intend to leave easily available reading material besidespublished articles and doctoral theses.

The readers are presented with the theory of control systems with sliding modeapplied to electrical motors and power converters. They can learn the methodology ofcontrol design and original algorithms of control and observation.

This book is important from a practical viewpoint, because nowadays practicallyall semiconductors devices are used in power electronics as power switches. Switchingpossesses myriad attractive inherent properties from a control viewpoint. Sliding mode

control systems supplies high dynamics to systems, invariability of systems to changesof parameters and exterior loads in the combination with simplicity of design. Unlikelinear control, switching sliding mode control does not replace the control system, butuses the natural properties of the control plant system effectively to ensure high controlquality.

There are very few available books on using sliding mode control for electricaldrives. Unlike other similar books, this book examines in detail the different featuresof various types of synchronous machines and converters from the viewpoint of slidingmode control design.

The manuscript presents a meticulous and detailed analysis of control issues andmechanical coordinate observation design for the various types of synchronousmachines and various drive control structures. The problem of drive parametersidentification is discussed as well.

The potential of the sliding mode control and observation is demonstrated innumerical and experimental results for real control plants.

The authors


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Acknowledgements and Dedications

To my son Denis

It is a pleasure for me to acknowledge the contributions of my former Universitylecturer Prof. V. Polkovnikov and my former Ph.D. supervisor Prof. V. Utkin, whomade me a scientist.

I wish to express my sincere gratitude to my colleagues at the Trapeznikov Instituteof Control Sciences of Russian Academy of Sciences and particularly to AcademicianProf. S. Vassilyev, Prof. A. Shubladze, Prof. V. Lotozky and Dr. D. Izosimov. Theirsupport was invaluable on the different writing up phases.

Thanks to Prof. Y. Rozanov, IEEE Fellow, for his important advice and remarkswhen preparing the final manuscript.

I would also like to express my sincerest thanks to Mrs. M. Platova. Without herassistance, this book could not have been started and completed.

Sergey Ryvkin

To my beloved parents Javier and Bertha

I would like to express a word of thanks to Dr. P.K. Sinha, for his invaluable help duringmy postgraduate studies, and to one of my former university lecturers in Mexico,Prof. Felipe Glvez, who made me a control engineer.

I also would like to express my deepest thankfulness to my adored parents, Javier ()and Bertha, an example of moral principles and behavior, hard work, dedication andlove, and to whom I owe everything.

Eduardo Palomar Lever

We would like this book to be free of errors, even if we know that this is impossible

in practical terms. So, we would appreciate if you could send any error reports to usto the following email addresses:[email protected],[email protected]


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About the authors

Sergey Ryvkinfirst graduated with high honors as an engineerfrom the Moscow Institute for Aviation Engineering (Techni-

cal University), after which he gained his PhD degree from theInstitute of Control Sciences (USSR Academy of Science) inMoscow and was awarded a DSc from the Supreme Certify-ing Commission of Russian Ministry of Education and Sciencein Moscow. He is currently a professor at the Russian StateUniversity for Humanities and a leading researcher at the Lab-oratory of Adaptive Control Systems for Dynamic objects atthe Trapeznikov Institute of Control Sciences from the RussianAcademy of Sciences. His lines of research are the application

of the sliding mode techniques to control of electrical drives and power systems andto their parameter observation. Prof. Ryvkin holds several patents and published onemonograph, five textbooks and more than 100 papers in international journals andproceedings. He is a member of the Russian Academy of Electrotechnical Sciences anda senior member of the IEEE.

Eduardo Palomar Lever got his BSc degree in electromechan-ical engineering from the National Autonomous Universityof Mexico, after which he obtained an MSc degree in con-

trol engineering and computing sciences from the Universityof Warwick, UK, and a PhD on sliding regimes to controlservomechanisms from the University of Sussex, UK. He isa full time research professor at the University of Guadala-jara. His lines of research focus on nonlinear control systemsand servomechanisms control using digital sliding modes.Prof. Palomar-Levers teaching specialties are advanced-levelcontrol engineering, automation, advanced mathematics, com-

puting languages, software development, and statistics. He earlier published a book

on ferroelectric materials as well as several papers on sliding motion control andbiomedicine in international journals and conference proceedings.
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2 Sliding mode control for synchronous electric drives

The first stage of development of relay system theory is connected first of all withthe names of H. Hazen (Hazen, 1934), A. Andronov (Andronov, 1959), Y. Tsypkin(Tsypkin, 1974) and I. Flugge-Lotz (Flugge-Lotz, 1953). Further from the relay systemtheory, the following directions were declared as independent:

The theory of nonlinear systems with various kinds of modulation (Kuerker, 2000),(Holz, 1994) and

The variable structure system theory (Emelyanov, 1970), (Edwards & Spurgeon,1998).

The founder of the last one is Prof. Stanislav Emelianov. He headed a variablestructure system academic school, whose scientists brought an essential contributionto this theory. The basic idea of this theory is the use of sliding modes in control

design. Sliding mode is a special kind of movement arising under certain conditions inswitching systems and inherent only to them. The specified mode provides in a dynamicsystem a high quality of control, invariance to external unmeasured disturbances, andsmall sensitivity to changes of dynamic properties of the plant.

The variable structure system theory was further developed and generalizedwith the theory of systems with discontinuous control (Utkin, 1978), (Utkin, 1992),(Edwards & Spurgeon, 1998), (Utkin et al, 1999), the theory of binary sys-tems (Emelyanov & Korovin, 2000) and the theory of higher order sliding modes(Fridman & Levant, 2002). The theory of systems with discontinuous control is basedon the use of a multidimensional sliding mode in the state space for the solution of con-

trol problems. The theory of binary systems is based on binary principles, i.e. the dualnature of signals in nonlinear dynamic systems that allows assigning operation designusing stabilizing feedback to an auxiliary nonlinear system. The theory of higher ordersliding modes generalizes the basic sliding mode acting on higher order time derivativesof the system, which totally removes the chattering effect and provides high accuracyin its implementation.

Possibility and perspectives of the use of sliding modes for alternating currentdrive control design was formulated for the first time in (Sabanovich & Izosimov,1981) though relay controllers found wide application in drive control before (Ilinsky,

2003). It is remarkable to note that, independently of the experts in the controlfield, the experts in the drive field also have addressed the use of relay control onthe basis of sliding modes (Brodovsky & Ivanov, 1974). These controls were usedin the phase current loops. Their use has been encouraged by the advances of newsemiconductor technologies and transition to power semiconductor voltage or currentconverters, whose power elements work in switching (relay) modes.

Rapid development of power semiconductor technologies has led to the deve-lopment of new types of high-frequency power devices, such as MOSFET and IGBT,which opened plenty of opportunities to create and perfect semiconductor power con-verters for drives (Benda, 1996), (Bose, 2002), (Mohan et al, 2003). All of the multi-dimensional relay control more actively used in the last decade worked mainly inphase current loops of the drive. The increasing number of publications testi-fies it. Such control in various publications has different names: relay control(Ilinsky, 2003), discontinuous control (Borcov, 1986), frequency-current control(Brodovky & Ivanov, 1974), sliding mode control (Utkin et al, 1999), (Ilinsky,


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Introduction 3

2003), (Cernat et al, 2000), (Vittek & Dodds, 2003), bang-bang control (Isidori,1999), hysteresis current control (Holz, 1994), (Holz & Beyer, 1994), currentforced control (Nagy, 1994), (Suetz et al, 1996), direct torque control (Leonard,2001), (Buja & Kazmierkowski, 2004), (Lascu et al, 2004), etc. Such variety of

names for one kind of control testifies that experts in the drive field do not have acommon viewpoint as to where and how to place the considered approach into theavailable control methods for the drives. The majority of the available publications aredevoted, as a rule, to disclosing private questions of research and implementation ofdrives with such control.

The term sliding mode control is preferred, in our opinion, because it providesthe best and fullest explanation of the processes involving the use of this kind ofcontrol. There is a whole theory of nonlinear systems with discontinuous controlsbehind this term. This theory widely explains not only the known high quality results

obtained by the use of this kind of control, but also those problems and complexitiesthat arise at its implementation.The difficulty of application of the sliding mode approach to the drive con-

trol design is that the methodology of the considered approach is highly theoretical.It uses mathematical models of the control plants that look like differential equa-tions with discontinuous control (Filippov, 1998), (Aizerman & Pyatnitskii, 1974).Besides that, direct use of the theory for drive control design is impossible withoutadditional research on questions concerning the organization of sliding modes, takinginto account specific features of the drive elements: electrical machines, semiconduc-tor power converters, sensors, etc. Though, as specified above, it is quite natural from

the viewpoint of the physical processes proceeding in the drive using the theory of sys-tems with discontinuous controls. Motor controls are voltages in the stator windingsof the electric machine. They have a discontinuous character, owing to switching kindof work of the semiconductor elements of the voltage converter. The discontinuouscharacter of the control, in this case, which is the defining sign of the theory of non-linear systems with discontinuous control, is not imposed to the system as an outsideproperty, but it is naturally defined by its physical nature.

The multidimensional relay characteristic of the power converter, which isdefined by the drive control design is not unique. It is necessary to consider the

nonlinearity of alternating current machines. In each of two alternating current electricmachines, i.e. asynchronous and synchronous, the process of transformation of electricpower into mechanical one has essential differences. It is caused by the basic distinctionin a source of a magnetic flux in the air gap, necessary for the creation of electromag-netic torque. The stator current generates this flux in the asynchronous motor usingthe electromagnetic induction. However, in the synchronous machine, it is generatedindependently by a flux source located in the rotor. Considering that the synchronousmachine combines very attractive properties as small rotor losses and good dynamicand accuracy characteristics, and in view of the fact that nonlinear characteristics ofthe former make an essential impact on drive control design, basic attention is givento problems of synchronous drive control design in the present monograph. The semi-conductor power converter and the synchronous motor enter into the drive structure.

Thus, a three-phase synchronous drive represents a nonlinear dynamic system withthe linear occurrence of the controlu(t) with discontinuous character, caused by theswitching operation mode of the elements of the power converter.


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The following prominent features of this class of nonlinear dynamic systemswith discontinuous control, in comparison with the widely researched traditional one(Utkin, 1978), (Utkin, 1992), are the following:

The amount of discontinuous control surpasses the dimension of control space(three-phase feed voltage of the electric machine at a two-dimensional voltagevector);

Basic vectors (orts) of the discontinuous controls used for the solution of a controlproblem are fixed;

Coefficients before the discontinuous controls are periodic (in the case of salient-pole (interior) synchronous motors).

Development of the theory of nonlinear systems with discontinuous controls to

such class of nonlinear systems has allowed developing design methods of nonlinearcontrol on sliding modes for the given concrete class of systems taking into accountits features, i.e. to physically use as much as possible potential possibilities for con-trol problem solving. With reference to three-phase drive, it means high quality ofcontrol processes, invariance to external perturbations, small sensitivity to changesof dynamic properties of plant, in a combination to profitability of power transmis-sion and simplicity of reception of the rotating magnetic field, inherent in three-phasecircuits.

Realization of the high-quality control based on use of a multidimensional slidingmode is impossible without due information support, which consists in reception of

the necessary information on state vector components of the plant. Direct measure-ment of all components of state vector needed for control design is impractical due toessential complications, cost issues and reduction of the plant operational reliability. Aperspective way of the solving of information support problem is working out of esti-mation algorithms for all the unmeasured components of the state vector based on theobserved components (Consoli, 2000), (Dote, 1988), (Krstic et al, 1995), (Luenberger,1966).

From the viewpoint of sliding mode the problem of the estimating algorithmsdesign has two parts. One part consists in receiving the state vector component esti-

mates needed for the sliding mode control design. Another part deals with using slidingmode techniques for receiving these estimates. In the latter case, methods of nonlinearobservation are based on the construction of a dynamic, imitating model of a nonlinearplant with discontinuous modeling control and the use of the attractive property ofsliding movement, which is the possibility of allocating an average continuous valueof a discontinuous control as a control information signal.

The present monograph contains sliding mode methods of control and observationdesign for nonlinear systems with a periodic matrix before redundant discontinuouscontrol, developed with uniform positions. Such approach allows making the mostof their structural features for achievement of the control goal. The offered approachwith reference to drives has allowed to develop design methods of the high-qualityinformation provided controls, both in continuous, and in discrete time. The synthe-sized controls fully use the physical nature of drive elements for a control problem.They are characterized by a high quality of control, invariance to external pertur-bations, small sensitivity to changes of feed voltage and dynamic properties of the


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Introduction 5

synchronous motor. They provide a high degree of usability of energy in a combina-tion to profitability of transmission of energy and simplicity of reception of the rotatingmagnetic field, inherent in the three-phase circuits.

The control tasks considered in this book found their reflection in the monograph

in content and structure, which consists of an introduction and eight chapters.Inchapter 1,the basic elements of the automated synchronous drive are classified

from a position of the automatic control theory as power converters and synchronousmotors. The mathematical descriptions of the power converters and the synchronousmotors are exposed. Control problems are formulated and formalized.

In section 1.1 classifications of the synchronous motors by a principle of a magneticflux creation and of the power converters by a principle of transformation of inputvoltage into a three-phase alternating voltage of the set frequency and amplitude arepresented. The mathematical models used for the control problem solving are shown.

In section 1.2, the basic requirements to drives are formulated and various drivesstructures and features of transformation of initial requirements depending on drivestructures are considered.

In chapter 2, theoretical backgrounds of multidimensional sliding movementdesign in nonlinear dynamic systems with a periodic matrix before redundant discon-tinuous control are stated. From uniform positions of the sliding mode theory controldesign problems are solved for a considered class of nonlinear systems. The sufficientexistence condition of a sliding mode is formulated and proved. This condition is thebase for a two step control design procedure.

In section 2.1, features of a considered class of the nonlinear dynamic systems are

analyzed. It is explained why the direct use of classical results of the theory of sys-tems with discontinuous controls for the concerned dynamics systems is not possible.Results of this classical sliding mode theory used in this work are produced.

In section 2.2, the sufficient existence conditions of a sliding mode in the systemsunder research are formulated and proved. The special case having a key importancefor the control design for the three-phase drives is considered.

A two-step procedure of a sliding mode control design for the investigated systemsis presented tosection 2.3.

Chapter 3is devoted to problems of information support of existence of multidi-

mensional sliding movement for nonlinear dynamic systems at a limited number of themeasured component of a state vector. Use of asymptotical observers, as informationbasis of the sliding movement design, providing invariance of sliding movement todiscrepancies of model and measurements in high frequency areas is proved. The solu-tion of a problem of nonlinear estimation state vector components by using slidingmode in a special nonlinear dynamic model with discontinuous controls is offered.

In section 3.1, information aspects of the organization of a sliding mode are dis-cussed. It is shown that from the information viewpoint it is possible to allocate twokinds of problems associated with sliding motion. First, the reception of the informa-tion on state variables, necessary for realization of sliding motion in a control loop.Second, the use of sliding motion for reception of the necessary information on statevariables.

In section 3.2, it is shown that the use of asymptotical observers is a methodolog-ical basis of a sliding mode design. Their use allows removing a problem of sensitivityof sliding movement in relation to high-frequency non-ideal behavior.
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Section 3.3 is devoted to working out problems of design methods of nonlinearobservers based on sliding modes for nonlinear systems with a linear occurrence of theestimated state vector components. Sufficient conditions of nonlinear estimation ofthe state vector components are formulated and proved. The observation algorithms

allow essentiallyreducing the demanded number of calculations.Chapter 4 is devoted to the problem of working out control design methods

and algorithms of automated synchronous drive with use of sliding modes for casesof one-loop and cascade control of drive and various types of synchronous motorsand power converters. A two-step decomposition design procedure of control allowedseparately considering by drive control design features of a synchronous motor anda supplied power converter is presented. Problems of maintenance of drive invari-ance to changes of plant parameters, external perturbations and changes of feedvoltage are discussed. Also, the problem of creating the reference value of a stator

current component idz based on the drive technical and economic requirements isdiscussed.In section 4.1, a decomposed two-step design method of a one-loop control is

described. On the first step of the control design only the features of the synchronousmotor in the rotating coordinates frame (d, q) is carried out. On the second step, thefeatures of the used power converter are considered and the phase voltage controls aresynthesized.

Section 4.2 presents the use of the above mentioned approach for design of thecascade control when the voltage source inverter works as a current source inverter.

Section 4.3 is devoted the solving of the optimization problems of drive power

characteristics by using the formation of the reference value of the stator current com-ponentidzin the closed loop. The reference value is formatted so that either it achievesmaximization of efficiency or it does not use reactive power.

Chapter 5is devoted to working out design methods of discontinuous control insystems with a multidimensional real sliding mode. The problems arising at use ofsliding modes for control in this case are analyzed. The controls providing a regu-larity commutating discontinuous component of control at the expense of a specialchoice of switching surfaces are synthesized. As an example, a current control prob-lem of the synchronous motor fed by the voltage source inverter is solved. The offered

approach allows providing high dynamic and accuracy of the system in a combina-tion with minimum switching losses and performance of electromagnetic compatibilityrequirements.

In section 5.1 the features of a real sliding mode caused by final frequency ofswitching of power devices and non-ideal of their relay characteristics are analyzed.Methods to regularize switching frequencies in a real sliding mode are planned.

In section 5.2 problems of optimization of feedforward (program) control by thevoltage source inverter on the basis of PWM, using switching losses minimization asa criterion, are solved. The optimal PWM for the voltage source inverter, based onallocation of zones of optimality, is synthesized with use of two step comparative ana-lyzes of possible PWMs by criterion of switching losses minimization. Implementationproblems are discussed.

In section 5.3 the method of control design providing realization of feedback PWMis developed, whose properties are equivalent to those of a feed forward PWM, i.e. theresulting real sliding mode is optimal on switching losses.
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Introduction 7

In section 5.4 the method of the control design providing regularization of thediscontinuous switching component of a control vector is developed.

Chapter 6isdevoted the estimation problems of drive output mechanical coor-dinates under the current information on drive electric variables. On the basis of

results of chapter 3, design methods of nonlinear state observers based on slidingmodes are offered. Estimation algorithms of mechanical coordinates for exterior syn-chronous motor and the synchronous reluctance one are designed. The features ofvarious observation algorithms are discussed. The block diagrams of nonlinear stateobservers are resulted.

In section 6.1 the general statement of an observation problem of drive mechanicalvariables is discussed. The features of a synchronous motor, as nonlinear plant areunderlined. It is emphasized that owing to specificity of these nonlinear plants, thedesign problem of an observation algorithm of mechanical coordinates must be solved

separately for each type of synchronous motor. Variables accessible to measurementare allocated.Section 6.2 presents the sliding mode observer for the exterior synchronous motor

with the permanent magnet excitation. The main idea is used as an observer a specialdynamics system, in which the sliding mode is willfully organized. The average valuesof the discontinuous controls give the information about the mechanical coordinates.Observation algorithms of variables in the stationer and rotating coordinates framesare produced.

In section 6.3 the above mentioned approach is used for the synchronous reluctancemotor. Observation algorithms are produced.

Chapter 7is devoted to design problems of a digital control for the synchronousdrive. Features of organization of digital control and a realization of a sliding modein such systems are considered. The design methods of digital control and observationfor synchronous drives, guaranteeing certainly step or asymptotical character of theprocesses are developed. Features of use of various control and observation are dis-cussed. For the purpose of simplification of control design procedure, the approachbased on use of the reference rate limiter is offered. It provides an exception of vari-able constrains in the course of system functioning. The design method of such limitersis developed. Conditions of system parameter identification that connect a depth of

memory, a frequency of quantization and a quantity of identified parameters are for-mulated. The problem of identification of the moment of inertia of the synchronousmotor is solved as an example. The condition of its identification is formulated. Prob-lems of the digital control of the drive with flexible joints are considered. The conditionof oscillatory free movements is formulated. The control for such drive is synthesized.

Features of digital control and organization of sliding movement are consideredin section 7.1.

Digital control design methods, based on difference equations, are developed insection 7.2.

The developed design methods of digital variable estimation under the retrospec-tive and current information are presented in section 7.3. The novel algorithms ofestimation and a filtration of drive output mechanical variables are developed.

The developed methods of design of digital algorithms of identification of systemparameters are presented in section 7.4. The algorithm of identification of the momentof inertia of a motor is developed.
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The developed design methods of digital algorithms of the reference rate limitationare presented in section 7.5. They provide an exception of mechanical coordinatesconstrains by the control design.

The developed design method of digital control of the drive with elastic mechanical

joints and the control synthesized with its use is presented in section 7.6.Chapter 8presents the results of the using the above supposed controls and obser-

vation algorithms for the control of the different technological processes. Features ofuse and technical realization of such control systems are discussed.

The solving design problem of digital control of the high speed synchronous drivewith a vector digital control without the mechanical coordinates sensors on a motorshaft is presented in section 8.1. The problem was a part of the federal target programcalled National technological base. The subject was Prototyping a small-sized highspeed drive for the oil drowned pumps for oil extracting with the inverter and micro-

processor control of power to 200 kW and the code of the subject was Nupor.(The directing agency is the federal state unitary enterprise Andronicus JosephianResearch and Production Enterprise All Russia Scientific Electro-Mechanics ResearchInstitute). The features of a solved problem imposing special requirements on controland approaches used are described. Modeling results have confirmed efficiency of ourapproach to digital control design.

Section 8.2 presents the digital control design for a drive with elastic mechanicaljoints, developed for the state unitary enterprise Instrument Design Bureau (StateUnitary Enterprise KBP). The name of developmental work was Prompting and stabi-lization drives of special plant. The code was Punzir, under the subject Working

out an alternating current drive with a vector and adaptive-modal microprocessorcontrol. The features of the solved problem and used approaches are described. Themodeling results have confirmed efficiency of our approach to designing digital control.

REFERENCES

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Dote, Y. Application of modern control techniques to motor control. Proc. of the IEEE, 1988,vol. 76, no. 4, pp. 438445.

Edwards, C. and Spurgeon, S.R. Sliding mode control: theory and applications. London:Taylor & Francis, 1998. 237 p.

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Chapter 1

Problem statement

1.1 MATHEMATICAL MODELS OF THE DRIVE ELEMENTS

1.1.1 Synchronous motors

Any electrical motor is a power transformer that converts electrical energy in mechan-ical one. When it is in action it works under the principle of interaction of currentsand magnetic fields, which leads to a general analytical description of the processesoccurring in any motor, including the present work. Because of the complexity ofthe electromagnetic and electromechanical processes happening in the real electricmachine, its full mathematical description is impractical, and from the control view-point it is ineffective due to the Dimension Curse, well known in classical control.

It is sufficient to use the classical mathematical models that consider the basic physicalfeatures of the processes happening inside the motor for control design (Leonhard,2001), (Boldea & Nasar, 2005). These mathematical descriptions are based on thefollowing standard assumptions:

The magnetomotive forces created by phase currents have sinusoidal distributionalong an air gap, i.e. influence of the higher frequency harmonics of a magneticfield is not considered;

Symmetry of the electrical machine; Influence of grooves is not considered, but the machine can be salient-pole;

Absence of saturation and losses in steel; The energy of any electrostatic field is not considered; Processes are concentrated.

In the frame of these assumptions the mathematical description of any electricmotor includes following three groups of equations:

Equations of electric balance in its windings; Equation of the electromagnetic torque developed by the electric motor;

Equation of mechanical movement (Newtons second law for a rotary motion).

The third equation, i.e. the equation of mechanical movement, is general for allelectric motors and looks like this:

Jd

dt=Mel M (1.1)


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whereJis the moment of inertia of all rotating parts, reduced to an motor rotor; isthe angular speed of the rotor;tis the current time;Melis the electromagnetic torquedeveloped by the electric motor;M = Ml(Fi) is the sum of the torques of the externalforces enclosed to a rotor.

The equations of electric balance based on Kirchhoffs second law, and the equa-tions of electromagnetic torque are defined by a kind of electric and magnetic circuitsof the electric engine and the physical processes happening inside it. Therefore, theyshould be considered independently for each type of synchronous motors.

From the control design viewpoint, all varieties of synchronous motors(Hanitsch & Parspour, 2000), (Pahman & Zhou, 2000) could be broken into severaltypes, depending on the following two criteria:

The rotor form, i.e. the form of an air gap in the electric engine.

The source of the magnetic flux causing the torque.

A classification of the types of synchronous motors based on such approach isshown infigure 1.1.

The first criterion in the given classification is the form of the air gap. There is auniform air gap in the case of a nonsalient-pole rotor. Otherwise there is a non-uniformgap between the rotor and the stator in a salient pole motor.

Figure 1.1 Classification of electric synchronous motors
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Problem statement 13

The magnetic flux in a synchronous motor can be created by permanent magnetsor a winding located on the rotor. In the latter case, sliding contacts are used to supplyelectric power to the rotor. The family of salient-pole synchronous motors concernsalso the synchronous reluctance machine, in which there is no independent source of

magnetic flux. Using the non-uniformity of the air gap forms the magnetic flux neededfor torque creation.

The mathematical description of the electromechanical transformation of powerin the synchronous motor using the actual phase currents and voltages as independentvariables gives a direct representation of the physical processes happening in the motor.However this mathematic model is complicated enough for analysis of the dynamicsprocesses. In the case of salient-pole machines this complexity is aggravated by factthat the synchronous motor description is a system of nonlinear differential equationswith periodic coefficients.

Parks equations are used in the theory of electrical machines for research on thesynchronous machine (Leonhard, 2001), (Boldea & Nasar, 2005), (Park, 1929). Theseequations are obtained by a linear Parks transformation of the above mentioned physi-cal equations of the synchronous motor. The result is a set of differential equations withfictitious variables in the rotating coordinates frame. (Adequacy of the mathematicaldescription is ensured using the condition of power invariance). This transformationprovides transition from the initial fixed coordinates system connected with motorphasesA, B, C to the rotating coordinates system (0, d, q) connected to a rotor. Thedatum line dis connected to the rotor flux, and the datum lineq is orthogonal to it. Thedatum line 0 is guided on an axis of rotation of the synchronous engine(figure 1.2).

The successful choice of transformation to axes (0, d, q) plays a fundamentalrole in the theory of synchronous machines and the theory of automatic drives, andclearly proves the following statements. First, thanks to the rotating coordinatessystem connected to a rotor, the differential equations describing the synchronousmotor dynamics have constant coefficients. Second, the orientation of a datum line0 on the axis of rotation makes the differential equation independent as regards azero sequence current. It must be emphasized that the zero sequence current doesnot participate in the magnetic field creation in an air gap, and, therefore, it doesnot have influence on the electromechanical processes in the motor, but only cre-

ates an additional loading of windings and semiconductor devices. The traditional

Figure 1.2 The synchronous motor


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connection of motor phase windings star or delta automatically provides equal-ity to zero of a zero sequence current, i.e. it eliminates additional thermal losses.From the viewpoint of the mathematical description of the synchronous motor itleads to its simplification. Its order is reduced after eliminating the differential

equation of the zero sequence current. In this case the considered three-phase syn-chronous motor is replaced by a two-phase idealized one (the generalized electricmachine), representing the collector machine, from the viewpoint of electromagneticprocesses.

The transformation from the three-phase coordinates system to the rotating two-phase one (d, q) is implemented by means of a Parks transformation matrix:

C=

2

3

cos A cosB cos C

sin A

sin B

sin C

(1.2)

where jis an electric angle between a rotor axis dand a stator phase axisj(j=A, B, C)and

2/3 is a power invariance factor.

The electric variables in a rotating coordinate system id, iq, ud, uq are also con-nected to phase variables:

I=CI, U=CU (1.3)

whereIT =(id, iq),IT =(iA, iB, iC),UT =(ud, uq),UT =(uA, uB, uC).The equations of electric balance and the electromagnetic moment for various

types of synchronous engines are shown below:

1.1.1.1 Salient-pole synchronous motor with an excitation winding

did

dt=

1

L21

(

Lfrid

+Ldfrfif

+LfLqiq)

+

Lf

L21

ud

Ldf

L21

uf,

diq

dt= 1

Lq(riq Ldid Ldfif) +

1

Lquq,

dif

dt= 1

L21(Ldrfif+ Ldfrid LdfLqiq) +

Ld

L21uf

Ldf

L21ud,

Mel=3

2p[Ldfif+ (Ld Lq)id]iq,

(1.4)

whereid,iq are the currents in the stator windings; if is the excitation current;ud,uq

are the voltages in the stator windings;ufis the voltage of the excitation winding;r isthe active resistance of a stator winding; rf is the active resistance of the excitationwinding; Ld, Lq are the inductance of stator windings; Lf is the inductance of theexcitation winding;Ldfis the mutual inductance excitation windings and stator wind-

ings d; L1=

LfLd L2df;p is the number of pair poles; is the electric angular speed.


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1.1.1.2 Permanent magnet salient-pole synchronous motor

diddt

= 1Ld

(rid+ Lqiq) +1

Ldud,

diqdt

= 1Lq

(riq Ldidf) + 1Lq

uq,

Mel=3

2p[f+ (Ld Lq)id]iq

(1.5)

wherefis a excitation flux.

1.1.1.3 Synchronous reluctance motor

diddt

= 1Ld

(rid+ Lqiq) +1

Ldud,

diq

dt= 1

Lq(riq Ldid) +

1

Lquq,

Mel=3

2p(Ld Lq)idiq

(1.6)

1.1.1.4 Nonsalient-pole synchronous motor with excitation winding

diddt

= 1L21

(Lfrid+ Ldfrfif+ LfLiq) +Lf

L21ud

Ldf

L21uf,

diq

dt= 1

L(riq Lid Ldfif) +

1

Luq,

dif

dt= 1

L21(Lrfif+ Ldfrid LdfLqiq) +

L

L21uf

Ldf

L21ud,

Mel=3

2pLdfifiq

(1.7)

whereLd=Lq=L are the inductances of the stator windings.

1.1.1.5 Permanent magnet nonsalient-pole synchronous motor

diddt

= 1L

(rid+ Liq) +1

Lud,

diq

dt= 1

L(riq Lidf) +

1

Luq,

Mel=3

2pfiq

(1.8)

1.1.2 Semiconductor power converters

To ensure the required operating mode of the synchronous motor, the correspondingvalues of feed voltages should be generated on its windings. However, as the sourceof power a supply-line with three-phase voltages (A, B, C) with fixed amplitude and


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Figure 1.3 Power converter block schemas. A) Direct transformation. B) With a direct current line

frequency usually acts. Nowadays, to transform these voltages into three-phase supplyvoltages for a synchronous motor (R,S,T) having a variable amplitude and frequencythree phase power converters are used, built using power transistors working in relaymode (Benda, 1996), (Bose, 2002), (Mohan & Underland, 2003).

The phase output voltage of the power converter represents in this case a sequenceof voltage impulses. This high frequency sequence is averaged in the synchronousmotor winding owing to its filtering property. The theoretical background is based onKotelnikovs or the sampling theorem (Kotelnikov, 1933), (Mark, 1991). This averagevoltage could be considered, as a continuous phase voltage on the synchronous motorwinding and it is a needed control to solve the control problem. Thus, the phase outputvoltage of the power converter from the control viewpoint has a dual character:

Pulse by nature; Continuous average to control the electric motor.

To realize the specified transformation of the supply line three-phase voltages,the scheme of direct transformation of electric power(figure 1.3.A),or the scheme ofpower transformation with a direct current link(figure 1.3.B)are used.

In the first case, the power converter directly produces the three-phase voltagesneeded for the synchronous motor control from the three-phase supply-line voltages.The power converter that performs such transformation is named a matrix converter.

In the second case the three-phase supply-line voltages is rectified by means of athree-phase rectifier and smoothed out by an output capacitor. The constant voltageobtained arrives on the input of the three-phase voltage source inverter (VSI). Fromthis DC line voltage the inverter produces the three-phase voltages needed for thesynchronous motor control.

Let us consider these two basic types of power converters from a controlviewpoint.


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Figure 1.5 VSI momentary output vectors

are not zero and also are the tops of the correct hexagon. Its symmetry axes are directedon the phase load orts.

The module of the above mentioned six nonzero vectors depends on the loadconnection circuit and the value of the input dc line voltageUin. In the load, connectedas star circuit, it is equal to 2Uin/3. The combinations of phase switches (pR,pS,pT)controls are presented near the possible vectors of the output voltageUi infigure 1.5.The zero vector corresponds to two combinations of the switch positions: either all

are connected to positive potential (111), or to negative potential (000).The VSI switch control, providing reception of the needed output voltage, includestwo independent parts (Zinoviev, 2005), (Ryvkin & Izosimov, 1997):

The modulation law that defines the part of the modulation period, in which thepower switch are connected either to the positive potential (Uin) or the negativeone (0);

The switching law that defines a sequence of switching of phase switches on themodulation period.

In this case, each phase output voltage represents a sequence of squared impulsesof various durations, whose amplitude is equal toUin, i.e. the feed voltage. A sequenceof these impulses, being averaged owing to filtering properties of the load, forms aphase output voltage on the load, which is a control tool. Thus, it is necessary toconsider the dual character of an output voltage vector by analysis of the VSI work


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Figure 1.6 Simplified schematic circuit of matrix converter

and control design. On one hand, it is characterized at any moment by the momentaryvalue, caused by momentary positions of the power switches. On the other hand,due to the averaging properties of the load, an average voltage value defines the VSIfeature as a component of the automatic control system. These problems will be alsoconsidered below by the control design of the synchronous drive.

Matrix converters (Bose, 2002) (Mohan, 2003). Progress in the area of high-voltage and frequency operated power devices opens new possibilities for the matrixconverter design. Primary benefits of such converters, which directly transform supply-line three-phase voltage (A, B, C) with constant frequency and amplitude into three-phase voltage (R, S, T) with desirable frequency and amplitude are:

Absence of reactive elements in the basic power scheme, i.e. the elements whichfilter high-frequency components of the output voltage and, hence, absence ofdynamic constrains;

Any direction of a power exchange between the load and the line.

The matrix converter demerits are the raised number of operated power devices(9 or 18, depending on the applied switch scheme) and rigid requirements to theswitching process. Either line short circuits or current disconnections of the load circuitduring switching are inadmissible.

The schematic circuit of the matrix converter is shown in figure 1.6. Theswitches KRA-KTC position depends on a control signalpij (pij {0, 1}, i=R, S, T;

j=A, B, C). The switches connect corresponding load phase (R,S,T) to correspond-ing supply-line phases (A, B, C). In this case the output voltage vectorUT =(U, U)


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in the fixed orthogonal coordinate system (, ) is defined as:

UU

=

2

31 1/2 1/2

0

3/2

3/2

URUS

UT

(1.10)

where numerical coefficients of a transformation matrix are directing the load phaseorts (R,S,T). The momentary output phase voltages of the matrix converter are equalto the corresponding supply line phase voltage. The sum of these phase voltages isequal to zero, so that the three-phase condition is satisfied. The supply-line phasevoltages form the three-phase system are:

UA=

Usint, UB=

Usint2

3, UC=

Usint+2

3 (1.11)

That is why it is possible at any moment to allocate a maximum (M), an interme-diate (k) and a minimum (m) values of voltage among them. They will be designatedaccordinglyUM,UkorUm. Since a primary goal of the matrix converter is the forma-tion of the demanded output there-phase voltage, it is more expedient to describe howit works, classifying three input voltages (1.11) not on a phase accessory, but on thesize of the input voltage, i.e. using the above mentioned maximumUM, intermediateUk and minimum Um voltage. We will enter a load phase control pi. This controlswitches three switches of a corresponding load phase and connects the load phase to

one of the supply-line three phases depending on its voltage value. The load phasecontrolpi can accept one of three values, pi {m, i, M}. Ifpi=m the load phase isconnected to a supply-line phase with the minimum value of the voltage, ifpi=k oneis connected to a supply-line phase with intermediate value of voltage, and ifpi=Mone is connected to a supply-line phase with the maximum value of the voltage. At anymoment each of load phases is connected to one of the supply-line phases. The loadphase control of each load phasepicould be transformed to the switch controlspijbyusing the information about the phase voltage value. As phase load control has a relaycharacter, the output voltage vectorUcan accept 25 values (figure 1.7): one of which

is zeroU0, and the others are 24 onesUl (l=1, . . . , 24) are non-zero.Figure 1.7 shows the momentary output voltage vector diagram of the matrix

converter containing three diagrams of the output voltage vectors that are typical forthe VSI. The tops of the six-vector set are the tops of the correct hexagons, whosesymmetry axes are directed on the directing load phase orts. The hexagon size, i.e. thevalue of the vector module, depends on which pair of the three input voltage valuesis being used, according to the momentary output three-phase voltage vectors and thetime.

The above mentioned hexagons are obtained when connected to a load with onlytwo of the three available input voltages. So the first hexagon is formed by vectorsU1, . . . , U6, where

U1=U1, U2=U1e j/3, U3=U1e j2/3,U4=U1e j, U5=U1e j4/3, U6=U1e j5/3


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Figure 1.7 Momentary output voltage vectors of matrix converter

These vectors are obtained by connecting only the maximum UMand minimum Uminput voltages to the load. The amplitude of the load voltage vectorU1 is not constantbut varies with the sextuple of the supply line frequency in the range from 1.5U to

3U.The second hexagon is formed by vectorsU7, . . . , U12, where

U7=U2, U8=U2e j/3, U9=U2e j2/3, U10=U2e j,U11

=U2e j4/3, U12

=U2e j5/3

These vectors are obtained by connecting to the load the intermediate Uk andminimum Um input voltages only. The load voltage vector amplitude U

2 is not aconstant but changes with the triple of the supply-line frequency in a range from1.5Uto 0.


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The third hexagon is formed by vectorsU13, . . . , U18, where

U13=

U3, U14=

U3e j/3, U15=

U3e j2/3,

U16=U3e j, U17=U3e j4/3, U18=U3e j5/3

These vectors are obtained by connecting to the load only the maximumUMandintermediate Ukinput voltages. The load voltage vector amplitude U

3 is not a constantbut changes with the triple of the supply-line frequency in a range from 1.5U to 0.The change of voltage vector amplitudes of the second and third hexagons occurs inanti-phase, i.e. if the voltage vector amplitude of one of them increases the amplitude

of another one decreases. The maximum moments of one of them coincide with theminimum moments of another.Besides, there are six vectorsU19, . . . , U24 of the constant amplitudeU

4 =1.5U,forming two three phase systems (U19, U20, U21), where U19=U4e jt, U20=U4e j(t2/3), U21=U4e j(t+2/3), and (U22, U23, U24), where U22=U4ejt, U23=U4ej(t2/3), U24=U4ej(t+2/3).

These two three-phase systems form a positive phase sequence and a negativeone. These systems rotate with the supply-line frequency and correspond to possibleconnections of the three-phase load to a three-phase supply-line.Figure 1.7presentsthe possible vectors of the output voltageUthat correspond to the combinations of

phase load controls (pR, pS, pT). A zero vector U0 corresponds to three combina-tions of the phase load controls: all three load phases are connected or to maximuminput phase voltage (MMM), or to intermediate one (kkk), or to minimum (mmm)one. A zero vector will have the top index value of the used voltages UM0 , U

k0 , U

m0

corresponding to each of these cases.The diagram of current vectors of the supply-line is similar to the diagram of load

voltage vectors. The differences are the following: instead of the load orts and themaximum, intermediate and minimum values of the supply-line voltages the supply-line ones and ones of load phase currents are respectively used. The current consumed

from a supply-line has the impulse form; the amplitude being equal to a load current.For current smoothing and decreasing interferences created by the matrix converter asupply-line filter could be applied. The needed values of the reactive filter elements ofthis filter are essentially less than the values of a choking inductance and capacity of aVSI DC-link, considered in the previous section.

It is possible, using modulation, to obtain the needed average values of the out-put voltages or the input currents.

Let us notice that there is another possibility to use the matrix switch scheme,similar to the matrix converter scheme to improve the quality of the converterpower consumption. The switch scheme could connect a line-supply and a DC-link. Thus, if the choking inductance is established at the input of such scheme,it is equivalent to the voltage source inverter considered earlier, and if the chokinginductance is established at the output, such schema is equal to the current inverter,respectively.


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1.2 DRIVE CONTROL PROBLEMS AND THEIR

EXISTING SOLUTIONS

An electrical motor in the drive structure carries out a transformation of the electric

energy into a mechanical one that moves the mechanisms participating in the workingprocess. The technology requirements for this process define the necessity and expe-diency of maintenance at the reference level of those or other mechanical variables,e.g.: position, speed, acceleration, torque etc, of the mechanism tip (Leonhard, 2001),(Boldea & Nasar, 2005), (Bose, 2002), (Ilinsky, 2003).

The main adjustable variable is usually a mechanical coordinate, more often arotor speed (t) or a rotor position (t), which should be equal to reference valuesz(t) (z(t)). The reference value z(t), in the most general case, is a time function.That is essentially the servo control problem. The actual rotor angular speed should

reproduce all changes of the reference with accuracy. Special cases of this problem are:

Stabilization of the rotor speed at the reference constant level; The rotor angular speed change under the reference law; Restricting the angular speed by an admissible value.

In relation to working off the reference demands, any change of the load torquehas to be carried out exactly and fast.

Together with the requirements on the mechanical coordinate control, demandson profitability of drive work are made. Its indicators are the efficiency coefficient,

characterizing power losses, and cos, characterizing consumption of the reactivepower.

Thus, the control goal is maintaining the reference value of the rotor velocity(t)in combination to power requirements performance. Since the real physical controlsare switch converter controls, defining a value and a sign of discontinuous (relay)control, the control problem consists in the formation of such switch controls so thatthe above mentioned requirements are fulfilled. Therefore, referring to the theory ofsystems with discontinuous controls is quite natural in this case.

Essential nonlinearities of both power converters and synchronous motors and

the complication of such drives bring up a variety of approaches to the solution of acontrol design problem.We can set apart several basic approaches to the design of synchronous drive

controls (Leonhard, 2001), (Pahman & Zhou, 2000), (Bose, 2002), (Ilinsky, 2003),for instance:

One-loop control; Decomposition one-loop control; Cascade (subordinated) control.

In the first case the drive is considered as a unit. The control design is interfacedto the solution of the difficult nonlinear problem. Its complexity is caused by essentialnonlinearities of the drive elements. The switching frequency and duration of switch-ing of various power switches are generated automatically in the closed loop, as an


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auxiliary element in the solution of the primary goal of drive control. Such dynamicsystems possess high dynamics and small sensitivity to drive parameter changesand external disturbances. Unfortunately, the automatically generated switching fre-quency of the power switches in this case is not a constant; it depends on the initial

conditions. This leads to power switches losses increase and to drive mechanical noise.In the second case two independent problems are considered, namely, the control

design of a synchronous motor and that of a power converter. The control for thesynchronous motor is synthesized on the assumption that the power converter gener-ates the synchronous motor input voltages responding to the solution of the primarygoal of control. The problem of the power converter control consists in transformingthe constant input voltage of an alternating or a direct current in three-phase voltagewith variable amplitude and frequency providing the general solution of the controlproblem.

This power converter control problem deals independently with the use of themodulation based on a high-frequency connection to the input voltages of the loadphases. In this case each phase output voltage of the power converter represents asequence of impulses of various durations, whose amplitude depends on the amplitudeof a momentary used input voltage. The sequence of these impulses, being averagedowing to the filtering feature of the load, forms a continuous phase output voltageon load, which is the motor control. In this case the indicators characterizing the dis-continuous control, i.e. its modulation, such as switching frequency and pulse ration,are formed for the power converter from the outside, i.e. feed forward (program) con-trol takes place. However, the automatic compensation of external disturbances and

internal parameter changes using such control is possible only in the closed loop of themechanical coordinate control.

The (subordinated) control approach is used by control design in the third case.It is based on the decomposition of an initial problem of the drive control on processesrates in the drive. The control problem for each of processes is handled independently.The control problem in the drive is divided by natural splitting of the processes: theelectromagnetic ones are fast and the mechanically ones are slow. Moreover, in relationto fast processes, mechanical variables are quasi constants. Hence the electromagneticprocesses control problem is reduced to a control of the electromagnetic torque. The

torque reference is formed in a slow control loop of the mechanical movement, whichis based on (1.1). Since the value of the electromagnetic torque is defined by the valuesthe currents in the synchronous motor windings, the control problem for a fast internalloop is reduced to a feedback current control using the power converter. The characterof change of discontinuous controls, i.e. the switching frequency and duration of theswitching of various power switches are generated automatically in the closed loop, asan auxiliary element of the solution of a current control problem. The current controlloop possesses high dynamics and small sensitivity to motor parameter changes andexternal disturbances. However, it is necessary to notice that, as well as in the firstcase, the switching frequency of power switches is generated automatically and itsvalue depends on initial conditions. It can lead to an increase of switching losses bythe power switches and mechanical drive noise.

Design problems of a high-quality drive are inseparably linked to problems ofreceiving the information of process state variables, in particular, about controlledcoordinates.


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The traditionally used approach based on direct measurement of all the necessarycoordinates, leads to considerable complication of the drive design, its operational dete-rioration and cost indexes. A possible way of overcoming these demerits is the exclusionof drive sensors for those coordinates whose direct measurement is undesirable, and the

control design using a state observer to obtain estimates (Leonhard, 2001), (Boldea &Nasar,), (Andreescu et al, 2000), (Dote, 1988).

The total elimination of mechanical coordinates sensors and design of a sensorlessdrive, containing only sensors of electric variables, is in perspective. The complexityof such approach is caused by the electromagnetic part of an electric motor, and inparticular the synchronous motor, as shown in section 1.1.1. It is described by thenonlinear equations (1.4)(1.8). Working out new nonlinear methods of estimation ofmechanical coordinates is necessary for reception on electric variables as estimationsof mechanical coordinates.

Features of control and observer design based on the theory of systems with dis-continuous controls for the drives used, the above presented types of synchronousmotors, and power converters will be considered in subsequent chapters. The newapproaches to drive control and observer design in continuous and discrete time, anddesigned with the help of novel algorithms to improve control quality are presented.

REFERENCES

Andreescu G.D., Popa A. and Spilca A. Sliding mode based observer for sensorless controlof PMSM drives two comparative study cases. Proc. the 7th International Conferenceon Optimization of Electrical and Electronical Equipment, OPTIM 2000, Brasov, Romania,2000, CD-ROM.

Benda V. Reliability of power semiconductor devices Problems and trends. Proc. 7th Inter-national Power Electronics & Motion Control Conference, PEMC96, Budapest, Hungary,1996, vol. 1, pp. 3035.

Boldea I. and Nasar S.A. Electric drives, 2nd ed. CRC Press, 2005. 544 p.Bose B.K. Modern power electronics and AC drives. New Jersey: Prentice Hall, 2002. 711 p.Consoli A. Advanced control techniques. Modern Electrical Drives. Dordrecht, Boston,

London: Kluwer Academic Publishers, 2000, pp. 523582.Dote Y. Application of modern control techniques to motor control. Proc. of the IEEE, 1988,

vol. 76, no. 4, pp. 438445.Hanitsch R. and Parspour N. Exterior Permanent Magnet Motors. Modern Electrical Drives.

Dordrecht, Boston, London: Kluwer Academic Publishers, 2000, pp. 79114.Holtz J. Sensorless control of induction machines with or without signal injection. Proc.

the 9th International Conference on Optimization of Electrical and Electronical Equipment,OPTIM 2004, Brasov, Romania, 2004, vol. II, pp. XVIIXXXIX.

Ilinsky N. Drive backgrounds. Moscow: Publishing house of Moscow power engineeringinstitute, 2003. 221 p. (in Russian).

Kotelnikov V.A. On the carrying capacity of the ether and wire in telecommunications, Mate-rial for the First All-Union Conference on Questions of Communication, Izd. Red. Upr. SvyaziRKKA, Moscow, 1933. (in Russian).

Leonhard W. Control of electrical drives. Berlin: Springer-Verlag, 2001. 460 p.Marks R.J. II: Introduction to Shannon sampling and interpolation theory. New York:

Spinger-Verlag, 1991. 332 p.


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Mohan N., Underland T.M. and Robbins W.P. Power electronics: converters, applications anddesign. 3rd edition. New York: John Wiley & Son Inc., 2003. 824 p.

Pahman M.A. and Zhou P. Interior permanent magnet motors. Modern Electrical Drives.Dordrecht, Boston, London: Kluwer Academic Publishers, 2000, pp. 115140.

Park R. Two-reaction theory of synchronous machines. AIEE Transactions, 1929, vol. 48,pp. 716730, 1933, vol. 52, pp. 352355.Ryvkin S.E. and Izosimov D.B. Comparison of pulse-width modulation algorithms for three-

phase voltage inverters. Electrical Technology, 1997, no. 2, pp. 133144.Zinoviev G.S. Power electronics backgrounds. Novosibirsk: Publishing house of Novosibirsk

state technical university, 2005. 664 p.


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The dynamic nonlinear systems with the sliding mode control possess very attractiveproperties, such as high quality of control, invariance to external perturbations, toler-ance to changes of plant parameters (Emelyanov et al, 1970), (Utkin, 1978), (Utkin,1992), (Utkin et al, 1999), (Zinober, 1994), (Edwards & Spurgeon, 1998).

However the prominent features of system listed above do not suppose direct useof known results of the theory of systems with sliding movements because knownresults of this theory are received, first of all, in the assumption that discontinuouscontrolsui(x, t) form the basis of a control vectoru(t), i.e. the discontinuous controlnumber is equal to the dimension of the control space, q = m.

It is therefore necessary to formulate the existence conditions of sliding modein the systems under research and to develop methods for the control design takinginto account the above mentioned features, the peculiar features of various controlschemes of three-phase electrical drive, and the various types used in them three-phase

semiconductor power converters and synchronous motors.The suggested approach to solve the above mentioned problems is based on break-

ing down an initial control problem into more simple ones using their step solution. Inthe first step, the sliding motion design in system (2.1) with a constant square matrixof a full rank B and discontinuous controls, forming the basis in the control space U, iscarried out. The design result is areas of admissible controls, ensuring the existence ofa sliding mode. In the second step, the coincidence problem for the areas of admissibleand realized controls according to (2.3) is solved.

The projection of a discontinuous control vectoru(t) to the control spaceU is

calculated according to the following transformation:

u=M(t)u (2.4)

whereM is a matrix of projection of discontinuous control vector on control space,whose dimension ism q.

Let us notice that in this case at each time in control spaceU there is a set ofrealized control vectorsU= {Ur},r=1, 2q.

It is known that in systems with discontinuous controls this specific kind of move-ment called sliding mode could appear. It is widely used to solve control problems

(Emelyanov et al, 1970), (Utkin, 1978), (Utkin, 1992), (Utkin et al, 1999), (Zinober,1994), (Edwards & Spurgeon, 1998). As it was marked above, such kind of motionpossesses a number of attractive properties: high quality of control, possibility ofinvariance to external unmeasured influences, small sensitivity to changes of dynamicproperties of the plant, and so on.

The basic question is under what conditions sliding motion exists in a discon-tinuous system. The problem of a sliding mode existence is equivalent to a stabilityproblem of an initial system (2.1)(2.4). It is solved in al-dimensional error space ofthe variablesZ(t)=zz(t) z(t), wherez(x, t) is the control variables vector,z(x)Rl,

andzz(x, t) is a reference of the control variables vector. In the theory of systems withdiscontinuous controls, zero errors of the components of control variablesZj(x, t)=0,j=1, l, or their linear transformations, are also named switching surfaces, since thesigns of discontinuous controls change on them. Using the terminology of the stabilitytheory for nonlinear systems, it is possible to speak about existence conditions of asliding mode in small and in big. The concept stability in small is equivalent


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Sliding mode in nonlinear dynamic systems 29

to a condition of existence of sliding movement on a switching surface or on cross-ing of such surfaces. Also, the concept of stability in big along with existence ofsliding movement on a switching surface or on crossing of such surfaces defines andthe reaching condition, i.e. a condition, which performance guarantees that a rep-resenting point from any initial position reaches a surface or crossing of surfaces ofswitching.

For the problem solving of the existence of sliding movement usually the methodof stability definition of Lyapunov (Emelyanov et al, 1970), (Utkin, 1978), (Utkin,1992), (Utkin et al, 1999), (Zinober, 1994), (Edwards & Spurgeon, 1998) is used.In this case, the equations of a projection of movement of initial system (2.1)(2.4) ona subspace of the control variables errors are analyzed:

dZ

dt=dzz

dt(Gf

+D

u) (2.5)

where G is a matrix of the dimension l n, which line are vectors-gradients of functionsZj(x, t); D=GBM.

Lemma: The dimension of the control variables vectorz(t) cannot outnumber a rankof any factor matrixes of matrixD, i.e.lmin[rank G, rank B(x, t), rank M].

In order to provide the control problem solving by using a classical sliding mode insystem (2.5) it is necessary that the dimension of an control variable error space or that

the same one of the control variable space would be less or equal to a projection of thecontrol space on this above mentioned ones. The projection dimension is defined by amatrix D rank. According to a consequence from the Binet-Cauchy formula the rank ofthis matrix cannot outnumbers the minimum rank of the factor matrixes. Thus, themaximum legitimate dimension of a control variable vectorz(t) is equal to a matrixDrank, i.e.lmin [rank G, rank B(x, t), rank M]. The statement is proved.

Let us notice that fact about the ranks of the factor matrixes. The matrix ofgradientsGgets out proceeding from a control problem in view and its rank, at desire,can be set as much as possible achievable, rank M=m, but the matrixB(x, t) rank is

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Sergey Ryvkin Eduardo Palomar Lever Sliding Mode Control for Synchronous Electric Drives 2012